{{Short description|Ordered field with a function generalizing the exponential function}} In mathematics, an '''ordered exponential field''' is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers.
==Definition== An exponential <math display="inline">E</math> on an ordered field <math display="inline">K</math> is a strictly increasing isomorphism of the additive group of <math display="inline">K</math> onto the multiplicative group of positive elements of <math display="inline">K</math>. The ordered field <math>K\,</math> together with the additional function <math>E\,</math> is called an ordered exponential field.
==Examples== * The canonical example for an ordered exponential field is the ordered field of real numbers '''R''' with any function of the form <math display="inline"> a^x</math> where <math display="inline">a</math> is a real number greater than 1. One such function is the usual exponential function, that is {{nowrap|1=''E''(''x'') = ''e''<sup>''x''</sup>}}. The ordered field '''R''' equipped with this function gives the ordered real exponential field, denoted by {{nowrap|1='''R'''<sub>exp</sub>}}. It was proved in the 1990s that '''R'''<sub>exp</sub> is model complete, a result known as Wilkie's theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that '''R'''<sub>exp</sub> is also o-minimal.<ref>A.J. Wilkie, ''Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function'', J. Amer. Math. Soc., '''9''' (1996), pp. 1051–1094.</ref> Alfred Tarski posed the question of the decidability of '''R'''<sub>exp</sub> and hence it is now known as Tarski's exponential function problem. It is known that if the real version of Schanuel's conjecture is true then '''R'''<sub>exp</sub> is decidable.<ref>A.J. Macintyre, A.J. Wilkie, ''On the decidability of the real exponential field'', Kreisel 70th Birthday Volume, (2005).</ref> * The ordered field of surreal numbers <math display="inline">\mathbf{No}</math> admits an exponential which extends the exponential function exp on '''R'''. Since <math display="inline">\mathbf{No}</math> does not have the Archimedean property, this is an example of a non-Archimedean ordered exponential field. * The ordered field of logarithmic-exponential transseries <math display="inline">\mathbb{T}^{LE}</math> is constructed specifically in a way such that it admits a canonical exponential.
==Formally exponential fields== A formally exponential field, also called an exponentially closed field, is an ordered field that can be equipped with an exponential <math display="inline">E</math>. For any formally exponential field <math display="inline">K</math>, one can choose an exponential <math display="inline">E</math> on <math display="inline">K</math> such that <math display="inline">1+1/n<E(1)<n</math> for some natural number <math display="inline">n</math>.<ref>Salma Kuhlmann, ''Ordered Exponential Fields'', Fields Institute Monographs, 12, (2000), p. 24.</ref>
==Properties== * Every ordered exponential field <math display="inline">K</math> is ''root-closed'', i.e., every positive element of <math>K\,</math> has an <math display="inline">n</math>-th root for all positive integers <math display="inline">n</math> (or in other words the multiplicative group of positive elements of <math>K\,</math> is divisible). This is so because <math display="inline">E\left(\frac{1}{n}E^{-1}(a)\right)^n=E(E^{-1}(a))=a</math> for all <math display="inline">a>0</math>. * Consequently, every ordered exponential field is a Euclidean field. * Consequently, every ordered exponential field is an ordered Pythagorean field. * Not every real-closed field is a formally exponential field, e.g., the field of real algebraic numbers does not admit an exponential. This is so because an exponential <math display="inline">E</math> has to be of the form <math>E(x)=a^x\,</math> for some <math display="inline"> 1<a\in K</math> in every formally exponential subfield <math display="inline">K</math> of the real numbers; however, <math>E(\sqrt{2})=a^\sqrt{2}</math> is not algebraic if <math display="inline"> 1<a </math> is algebraic by the Gelfond–Schneider theorem. * Consequently, the class of formally exponential fields is not an elementary class since the field of real numbers and the field of real algebraic numbers are elementarily equivalent structures. * The class of formally exponential fields is a pseudoelementary class. This is so since a field <math>K\,</math> is exponentially closed if and only if there is a surjective function <math display="inline">E_2\colon K\rightarrow K^+</math> such that <math display="inline">E_2(x+y)=E_2(x)E_2(y)</math> and <math display="inline">E_2(1)=2</math>; and these properties of <math display="inline">E_2</math> are axiomatizable.
==See also== * Exponential field
==Notes== {{Reflist}}
==References== * {{cite journal | first=Norman L. | last=Alling | title=On Exponentially Closed Fields | journal=Proceedings of the American Mathematical Society | volume=13 | number=5 | year=1962 | pages=706–711 | zbl=0136.32201 | doi=10.2307/2034159| jstor=2034159 | doi-access=free }} * {{Citation|last=Kuhlmann|first=Salma|title=Ordered Exponential Fields|series=Fields Institute Monographs|volume=12|publisher=American Mathematical Society|year=2000|isbn=0-8218-0943-1|mr=1760173|doi=10.1090/fim/012|doi-access=free}}
Category:Model theory Category:Field theory Category:Algebraic structures Category:Exponentials